Why does this equation hold for $S > S_f$?
Could you give me link for proof?

Another question is Why do we need high-contact condition?

UPD. Do I correctly understand that for American Put Option
If $S > S_{f}$, there is no sence to exercise at time $t<T$ (because it causes immediate loss: $-V+S-K<0$).
So it behaves like European Option, hence $V^{Am}_{P}=V^{E}_{P}$ and it satisfies Black-Scholes Eqation.

now assume there is a $V(S,t)<\max(K-S(t),0)$: there would be the opportunity for arbitrage. We could buy the asset for $S$ and the put option for $V$. Selling the asset for $K$ would lead to a risk free profit of $K-S-V$. Thus the value of the american put option must hold the additional constraint
$$V(S,t)\geq \max(K-S(t),0)$$

As long as $V > K-S$ (or $S>S_f$) it is given by the BS PDE, otherwise the price is given by $K-S$. Most (if not all) textbook introductions to financial derivatives include more details on that and derivations of the BS PDE.

The second (more mysterious) constraint is a consequence of "optimal" behavior of the agents. Keywords to find more on that might be optimal stopping problem or game theory of options.